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Let $F=1-2M/r$, define the tortoise coordinate by $d{r}_{\ast }/dr=F^{-1}$, and set the advanced time $v=t+r_{\ast }$. Since $dt=dv-F^{-1}dr$, substitution givesThese are Ingoing Eddington-Finkelstein coordinates; all coefficients are regular at the event horizon $r=2M$.

Outside the ingoing null shell the enclosed mass is $E$, so use the ingoing Ingoing Eddington-Finkelstein coordinatesThis is the Schwarzschild metric of mass $E$, regular across $r=2E$ and valid for $r>0$ on the exterior side of the shell.

Define the ingoing coordinateThen $dt=dv-dr/f$ andThis is the higher-dimensional analogue of Ingoing Eddington-Finkelstein coordinates. Its metric and inverse are regular at $r=r_{+}$, including when the zero is degenerate. A gauge transformation removes the singular radial term in the transformed potential, leaving the regular representative $A=-Q{r}^{-(d-3)}dv$. These expressions analytically extend the spacetime through the future outer horizon.

Since $v=t+r_{\ast }$ and $d{r}_{\ast }=f^{-1}dr$,Substitution into the metric givesThese are Ingoing Eddington-Finkelstein coordinates. Every coefficient is finite at $r=2M$, and the determinant of the $v,r$ block is $-1$. Thus the apparent singularity there in Schwarzschild coordinates is a coordinate singularity; the metric extends smoothly across the Schwarzschild event horizon.